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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 0
[1,2] => [[1,2]]
=> [1,2] => 1
[2,1] => [[1],[2]]
=> [2,1] => 0
[1,2,3] => [[1,2,3]]
=> [1,2,3] => 2
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => 1
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 3
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,4,1,3] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,2,4,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
Description
The number of ascents of a permutation.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> [1]
=> 0
[1,2] => [1,1]
=> [2]
=> [1,1]
=> 1
[2,1] => [2]
=> [1,1]
=> [2]
=> 0
[1,2,3] => [1,1,1]
=> [3]
=> [1,1,1]
=> 2
[1,3,2] => [2,1]
=> [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [3]
=> 1
[2,3,1] => [2,1]
=> [2,1]
=> [3]
=> 1
[3,1,2] => [2,1]
=> [2,1]
=> [3]
=> 1
[3,2,1] => [3]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> [4]
=> 2
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,3,2,1] => [4]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [5]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [3,2]
=> [5]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1
[8,7,6,2,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[7,6,5,1,4,3,8,2] => ?
=> ?
=> ?
=> ? = 2
[9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ?
=> ? = 1
[9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ?
=> ? = 1
[10,9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ?
=> ? = 1
[10,9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ?
=> ? = 1
[10,9,7,6,5,4,3,2,1,8] => ?
=> ?
=> ?
=> ? = 1
[4,3,1,5,7,8,2,6] => ?
=> ?
=> ?
=> ? = 5
[6,7,2,1,8,4,3,5] => ?
=> ?
=> ?
=> ? = 5
[4,3,1,6,7,8,2,5] => ?
=> ?
=> ?
=> ? = 5
[2,3,4,5,7,8,9,6,1] => ?
=> ?
=> ?
=> ? = 6
[2,4,5,6,7,8,9,3,1] => ?
=> ?
=> ?
=> ? = 6
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [1,1]
=> 1
[2,1] => [2]
=> 0
[1,2,3] => [1,1,1]
=> 2
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [2,1]
=> 1
[3,1,2] => [2,1]
=> 1
[3,2,1] => [3]
=> 0
[1,2,3,4] => [1,1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [2,1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> 2
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [3,1]
=> 1
[3,2,4,1] => [3,1]
=> 1
[3,4,1,2] => [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> 1
[4,1,2,3] => [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [3,1]
=> 1
[4,3,1,2] => [3,1]
=> 1
[4,3,2,1] => [4]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> 3
[5,6,4,3,7,8,1,2] => ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ? = 6
[3,2,5,8,1,4,6,7] => ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ? = 2
[] => []
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ? = 1
[8,7,6,2,5,3,4,1] => ?
=> ? = 2
[7,6,5,1,4,3,8,2] => ?
=> ? = 2
[9,8,7,6,5,3,2,1,4] => ?
=> ? = 1
[9,8,7,5,4,3,2,1,6] => ?
=> ? = 1
[10,9,8,7,6,5,3,2,1,4] => ?
=> ? = 1
[10,9,8,7,5,4,3,2,1,6] => ?
=> ? = 1
[10,9,7,6,5,4,3,2,1,8] => ?
=> ? = 1
[4,3,1,5,7,8,2,6] => ?
=> ? = 5
[6,7,2,1,8,4,3,5] => ?
=> ? = 5
[4,3,1,6,7,8,2,5] => ?
=> ? = 5
[2,3,4,5,7,8,9,6,1] => ?
=> ? = 6
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ? = 6
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ? = 2
[] => []
=> ?
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ? = 1
[8,7,6,2,5,3,4,1] => ?
=> ?
=> ? = 2
[7,6,5,1,4,3,8,2] => ?
=> ?
=> ? = 2
[9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ? = 1
[9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ? = 1
[10,9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ? = 1
[10,9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ? = 1
[10,9,7,6,5,4,3,2,1,8] => ?
=> ?
=> ? = 1
[4,3,1,5,7,8,2,6] => ?
=> ?
=> ? = 5
[6,7,2,1,8,4,3,5] => ?
=> ?
=> ? = 5
[4,3,1,6,7,8,2,5] => ?
=> ?
=> ? = 5
[2,3,4,5,7,8,9,6,1] => ?
=> ?
=> ? = 6
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001034
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 1
[2,1] => [2]
=> []
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,3,1] => [2,1]
=> [1]
=> [1,0]
=> 1
[3,1,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[3,2,1] => [3]
=> []
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,3,2,1] => [4]
=> []
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2
[] => []
=> ?
=> ?
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1
[8,7,6,2,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[7,6,5,1,4,3,8,2] => ?
=> ?
=> ?
=> ? = 2
[9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ?
=> ? = 1
[9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ?
=> ? = 1
[10,9,8,7,6,5,3,2,1,4] => ?
=> ?
=> ?
=> ? = 1
[10,9,8,7,5,4,3,2,1,6] => ?
=> ?
=> ?
=> ? = 1
[10,9,7,6,5,4,3,2,1,8] => ?
=> ?
=> ?
=> ? = 1
[4,3,1,5,7,8,2,6] => ?
=> ?
=> ?
=> ? = 5
[6,7,2,1,8,4,3,5] => ?
=> ?
=> ?
=> ? = 5
[4,3,1,6,7,8,2,5] => ?
=> ?
=> ?
=> ? = 5
[2,3,4,5,7,8,9,6,1] => ?
=> ?
=> ?
=> ? = 6
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000293
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> => ? = 0
[1,2] => [1,1]
=> [1]
=> 10 => 1
[2,1] => [2]
=> []
=> => ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 110 => 2
[1,3,2] => [2,1]
=> [1]
=> 10 => 1
[2,1,3] => [2,1]
=> [1]
=> 10 => 1
[2,3,1] => [2,1]
=> [1]
=> 10 => 1
[3,1,2] => [2,1]
=> [1]
=> 10 => 1
[3,2,1] => [3]
=> []
=> => ? = 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,4,3,2] => [3,1]
=> [1]
=> 10 => 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,1,4,3] => [2,2]
=> [2]
=> 100 => 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,4,3,1] => [3,1]
=> [1]
=> 10 => 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,1,4,2] => [2,2]
=> [2]
=> 100 => 2
[3,2,1,4] => [3,1]
=> [1]
=> 10 => 1
[3,2,4,1] => [3,1]
=> [1]
=> 10 => 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,4,2,1] => [3,1]
=> [1]
=> 10 => 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,1,3,2] => [3,1]
=> [1]
=> 10 => 1
[4,2,1,3] => [3,1]
=> [1]
=> 10 => 1
[4,2,3,1] => [3,1]
=> [1]
=> 10 => 1
[4,3,1,2] => [3,1]
=> [1]
=> 10 => 1
[4,3,2,1] => [4]
=> []
=> => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1010 => 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 1010 => 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 110 => 2
[5,4,3,2,1] => [5]
=> []
=> => ? = 0
[6,5,4,3,2,1] => [6]
=> []
=> => ? = 0
[7,6,5,4,3,2,1] => [7]
=> []
=> => ? = 0
[8,7,6,5,4,3,2,1] => [8]
=> []
=> => ? = 0
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ? => ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ? => ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ? => ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ? => ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ? => ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ? => ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ? => ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ? => ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ? => ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ? => ? = 6
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> => ? = 0
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ? => ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ? => ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ? => ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ? => ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ? => ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ? => ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ? => ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ? => ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ? => ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ? => ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ? => ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ? => ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ? => ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ? => ? = 2
[] => []
=> ?
=> ? => ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> => ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> => ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ? => ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ? => ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ? => ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ? => ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ? => ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ? => ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ? => ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ? => ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ? => ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ? => ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ? => ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ? => ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ? => ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ? => ? = 1
Description
The number of inversions of a binary word.
Matching statistic: St000459
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> ?
=> ? = 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> ?
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,4,3,2,1] => [5]
=> []
=> ?
=> ? = 0
[6,5,4,3,2,1] => [6]
=> []
=> ?
=> ? = 0
[7,6,5,4,3,2,1] => [7]
=> []
=> ?
=> ? = 0
[8,7,6,5,4,3,2,1] => [8]
=> []
=> ?
=> ? = 0
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> ?
=> ? = 0
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2
[] => []
=> ?
=> ?
=> ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> ?
=> ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> ?
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> ?
=> ? = 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> ?
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,4,3,2,1] => [5]
=> []
=> ?
=> ? = 0
[6,5,4,3,2,1] => [6]
=> []
=> ?
=> ? = 0
[7,6,5,4,3,2,1] => [7]
=> []
=> ?
=> ? = 0
[8,7,6,5,4,3,2,1] => [8]
=> []
=> ?
=> ? = 0
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> ?
=> ? = 0
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2
[] => []
=> ?
=> ?
=> ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> ?
=> ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> ?
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> ?
=> ? = 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> ?
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,4,3,2,1] => [5]
=> []
=> ?
=> ? = 0
[6,5,4,3,2,1] => [6]
=> []
=> ?
=> ? = 0
[7,6,5,4,3,2,1] => [7]
=> []
=> ?
=> ? = 0
[8,7,6,5,4,3,2,1] => [8]
=> []
=> ?
=> ? = 0
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> ?
=> ? = 0
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2
[] => []
=> ?
=> ?
=> ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> ?
=> ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> ?
=> ? = 0
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001382
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> ?
=> ? = 0 - 1
[1,2] => [1,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1] => [2]
=> []
=> ?
=> ? = 0 - 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,2,1] => [3]
=> []
=> ?
=> ? = 0 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,3,2] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[3,1,4,2] => [2,2]
=> [2]
=> [2]
=> 1 = 2 - 1
[3,2,1,4] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[3,4,2,1] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,2,3,1] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,3,2,1] => [4]
=> []
=> ?
=> ? = 0 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1 = 2 - 1
[5,4,3,2,1] => [5]
=> []
=> ?
=> ? = 0 - 1
[6,5,4,3,2,1] => [6]
=> []
=> ?
=> ? = 0 - 1
[7,6,5,4,3,2,1] => [7]
=> []
=> ?
=> ? = 0 - 1
[8,7,6,5,4,3,2,1] => [8]
=> []
=> ?
=> ? = 0 - 1
[5,6,4,3,7,8,1,2] => ?
=> ?
=> ?
=> ? = 4 - 1
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ?
=> ? = 3 - 1
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ?
=> ? = 3 - 1
[2,3,4,7,8,6,5,1] => ?
=> ?
=> ?
=> ? = 4 - 1
[1,2,5,8,7,6,4,3] => ?
=> ?
=> ?
=> ? = 3 - 1
[1,2,4,7,8,6,5,3] => ?
=> ?
=> ?
=> ? = 4 - 1
[2,1,3,5,6,8,4,7] => ?
=> ?
=> ?
=> ? = 6 - 1
[2,6,7,1,3,8,4,5] => ?
=> ?
=> ?
=> ? = 6 - 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 6 - 1
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ?
=> ? = 6 - 1
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> ?
=> ? = 0 - 1
[3,2,5,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 5 - 1
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 5 - 1
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 5 - 1
[6,2,5,8,1,3,4,7] => ?
=> ?
=> ?
=> ? = 5 - 1
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 5 - 1
[8,2,5,7,1,3,4,6] => ?
=> ?
=> ?
=> ? = 5 - 1
[6,8,3,7,1,2,4,5] => ?
=> ?
=> ?
=> ? = 5 - 1
[5,4,3,7,1,2,6,8] => ?
=> ?
=> ?
=> ? = 4 - 1
[5,3,2,8,1,4,6,7] => ?
=> ?
=> ?
=> ? = 4 - 1
[5,4,3,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 4 - 1
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 4 - 1
[8,3,2,5,7,1,4,6] => ?
=> ?
=> ?
=> ? = 4 - 1
[6,4,7,3,5,1,2,8] => ?
=> ?
=> ?
=> ? = 4 - 1
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 2 - 1
[] => []
=> ?
=> ?
=> ? = 0 - 1
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> ?
=> ? = 0 - 1
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> ?
=> ? = 0 - 1
[3,5,4,2,6,8,1,7] => ?
=> ?
=> ?
=> ? = 4 - 1
[2,5,7,8,1,6,4,3] => ?
=> ?
=> ?
=> ? = 4 - 1
[1,5,7,8,2,6,4,3] => ?
=> ?
=> ?
=> ? = 4 - 1
[1,8,9,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[1,7,9,8,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[1,9,10,8,7,6,5,4,3,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[1,4,5,8,3,7,6,2] => ?
=> ?
=> ?
=> ? = 4 - 1
[8,2,7,6,5,3,4,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,2,7,5,4,3,6,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[2,8,7,6,4,3,5,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,4,7,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,6,3,5,4,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,1,7,3,6,5,4,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,6,2,5,4,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000738The first entry in the last row of a standard tableau. St000734The last entry in the first row of a standard tableau. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000074The number of special entries. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000369The dinv deficit of a Dyck path. St000839The largest opener of a set partition. St001726The number of visible inversions of a permutation. St000502The number of successions of a set partitions. St000211The rank of the set partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000362The size of a minimal vertex cover of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000354The number of recoils of a permutation. St000021The number of descents of a permutation. St000209Maximum difference of elements in cycles. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice.
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